In Exercises 5–12, graph two periods of the given tangent function. y = −2 tan 1/2 x

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1. Identify the standard form of the tangent function: $y=a\tan (bx-c)+d$. In this case, $a=-2$, $b=\frac{1}{2}$, $c=0$, and $d=0$.
2. Determine the period of the tangent function. The period of $\tan (bx)$ is $\frac{\pi }{b}$. Substitute $b=\frac{1}{2}$ to find the period: $\frac{\pi }{\frac{1}{2}}=2\pi$.
3. Identify the vertical stretch and reflection. The coefficient $a=-2$ indicates a vertical stretch by a factor of 2 and a reflection across the x-axis.
4. Determine the asymptotes. For $\tan (bx)$, the vertical asymptotes occur at $bx=\frac{\pi }{2}+k\pi$, where $k$ is an integer. Solve for $x$ to find the asymptotes: $\frac{1}{2}x=\frac{\pi }{2}+k\pi$, which simplifies to $x=\pi +2k\pi$.
5. Graph two periods of the function. Start from the first asymptote, plot the key points (midpoint and quarter points), and sketch the curve, considering the vertical stretch and reflection.

Key Concepts

Tangent Function

The tangent function, denoted as tan(x), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It is periodic with a period of π, meaning it repeats its values every π radians. Understanding the properties of the tangent function, including its asymptotes and behavior, is essential for graphing it accurately.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. In the given equation, y = -2 tan(1/2 x), the coefficient -2 indicates a vertical reflection and a vertical stretch by a factor of 2, while the 1/2 inside the tangent function indicates a horizontal stretch, effectively doubling the period of the function to 2π.

Graphing Trigonometric Functions

Graphing trigonometric functions requires understanding their key features, such as amplitude, period, phase shift, and vertical shift. For the tangent function, identifying the asymptotes, which occur where the function is undefined, is crucial. In this case, the graph will have vertical asymptotes at x = (2n + 1)π/2, where n is an integer, and the graph will repeat every 2π due to the horizontal stretch.